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AIP Advances

Probabilistic quantum cloning of three symmetric states

Shouya Wang, Wen‐Hai Zhang

Published Dec 1, 2024• 0 citations
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Abstract

This paper investigates the 1 → M probabilistic quantum cloning of three symmetric states, which are defined by a complex number of their inner product. First, a general transformation of the 1 → M probabilistic quantum cloning is designed, and the ancillary success states are introduced. Then, the explicit expression of the probability of successful cloning is derived, which depends on the number M and the inner product of the three symmetric states. Finally, three examples are provided to demonstrate the efficiency of our cloning transformation.

DOI
10.1063/5.0228577

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This paper investigates the 1 → M probabilistic quantum cloning of three symmetric states, which are defined by a complex number of their inner product. First, a general transformation of the 1 → M probabilistic quantum cloning is designed, and the ancillary success states are introduced. Then, the explicit expression of the probability of successful cloning is derived, which depends on the number M and the inner product of the three symmetric states. Finally, three examples are provided to demonstrate the efficiency of our cloning transformation.
We study the probabilistic cloning of three symmetric states. These states are defined by a single complex quantity, the inner product among them. We show that three different probabilistic cloning machines are necessary to optimally clone all possible families of three symmetric states. We also show that the optimal cloning probability of generating MMM copies out of one original can be cast as the quotient between the success probability of unambiguously discriminating one and MMM copies of symmetric states.
We study the probabilistic cloning of equidistant states. These states are such that the inner product between them is a complex constant or its conjugate. Thereby, it is possible to study their cloning in a simple way. In particular, we are interested in the behavior of the cloning probability as a function of the phase of the overlap among the involved states. We show that for certain families of equidistant states Duan and Guo's cloning machine leads to cloning probabilities lower than the optimal unambiguous discrimination probability of equidistant states. We propose an alternative cloning machine whose cloning probability is higher than or equal to the optimal unambiguous discrimination probability for any family of equidistant states. Both machines achieve the same probability for equidistant states whose inner product is a positive real number.
We construct a probabilistic cloning and deleting machine which, taking several copies of an input quantum state, can output a linear superposition of multiple cloning and deleting states. Since the machine can perform cloning and deleting in a single unitary evolution, the probabilistic cloning and other cloning machines proposed in the previous literature can be thought of as special cases of our machine. A sufficient and necessary condition for successful cloning and deleting is presented, and it requires that the copies of an arbitrarily presumed number of the input states are linearly independent. This simply generalizes some results for cloning. We also derive an upper bound for the success probability of the cloning and deleting machine.
Two known nonorthogonal quantum states can be cloned either deterministically or probabilistically. In this paper, we investigate quantum cloning by combining these two extreme cases, i.e., a trade-off between the copy fidelity and the success probability. For a special set of two known nonorthogonal quantum states, we start with an explicit unitary transformation and derive the copy fidelity as a function of the success probability. The result shows that the higher is the copy fidelity, the less is the success probability. This quantum cloning may have important applications in quantum cryptography.